| L(s) = 1 | − 6·9-s + 6·25-s − 24·37-s + 14·49-s + 96·61-s − 48·73-s + 9·81-s − 56·109-s − 42·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 56·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2·9-s + 6/5·25-s − 3.94·37-s + 2·49-s + 12.2·61-s − 5.61·73-s + 81-s − 5.36·109-s − 3.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.013330709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.013330709\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 - 3 T^{2} - 16 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 6 T^{2} - 253 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 10 T^{2} - 261 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 42 T^{2} + 1235 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - T^{2} - 960 T^{4} - p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + p T^{2} )^{8} \) |
| 47 | \( ( 1 - 46 T^{2} - 93 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 85 T^{2} + 4416 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 29 T^{2} - 2640 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 24 T + 253 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 50 T^{2} - 1989 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 137 T^{2} + 12528 T^{4} - 137 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 163 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 150 T^{2} + 14579 T^{4} - 150 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.37600914229892118006587994266, −4.29952061084074496589446337564, −4.28510877062922424604465617127, −4.12558414617889385293933136261, −4.09155892205019181368837896097, −3.83922101874522064698186962343, −3.73694966527679362982677836755, −3.60421863314434310000569042147, −3.48115644035705421958308767734, −3.35488371896410200407835302408, −3.06899463931710741948924404006, −2.95086395418404648738431891695, −2.79451010913874042032017751818, −2.64307183575891835413264397995, −2.54912301139708122415280824491, −2.44292290300419857843594206698, −2.26690303730499291923912147854, −1.92594207516921656527337159038, −1.87286288056777441256106998321, −1.71927773174676354592339269092, −1.25317789646237664540496918782, −1.10699186861306373941749029936, −0.958659548847434545362513805522, −0.46451106891758002177479458560, −0.29863038665977127894126369644,
0.29863038665977127894126369644, 0.46451106891758002177479458560, 0.958659548847434545362513805522, 1.10699186861306373941749029936, 1.25317789646237664540496918782, 1.71927773174676354592339269092, 1.87286288056777441256106998321, 1.92594207516921656527337159038, 2.26690303730499291923912147854, 2.44292290300419857843594206698, 2.54912301139708122415280824491, 2.64307183575891835413264397995, 2.79451010913874042032017751818, 2.95086395418404648738431891695, 3.06899463931710741948924404006, 3.35488371896410200407835302408, 3.48115644035705421958308767734, 3.60421863314434310000569042147, 3.73694966527679362982677836755, 3.83922101874522064698186962343, 4.09155892205019181368837896097, 4.12558414617889385293933136261, 4.28510877062922424604465617127, 4.29952061084074496589446337564, 4.37600914229892118006587994266
Plot not available for L-functions of degree greater than 10.
